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THE STEINBERG REPRESENTATION Introduction. Group Representations Occupy a Sort of Middle Ground Between Abstract Groups and Tran BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 16, Number 2, April 1987 THE STEINBERG REPRESENTATION J. E. HUMPHREYS To Robert Steinberg on his 65th birthday Introduction. Group representations occupy a sort of middle ground between abstract groups and transformation groups, i.e., groups acting in concrete ways as permutations of sets, homeomorphisms of topological spaces, diffeomor- phisms of manifolds, etc. The requirement that the elements of a group act as linear operators on a vector space limits somewhat the complexity of the action without sacrificing the depth or applicability of the resulting theory. As in other areas of mathematics, study of linear phenomena may illuminate more general phenomena. The widespread use of group representations in mathematics (as well as in physics, chemistry,... ) does not imply the existence of a single unified subject, however. Nor do practitioners always understand one another's language. Groups come in many flavors: finite, infinite-but-discrete, compact, locally compact, etc. Vector spaces may be finite or infinite dimensional; in the latter case there might be a Hilbert space structure and operators might be required to be unitary. The underlying scalar field may be complex, real, /?-adic, finite,.... One can also make groups act on free modules over rings of arithmetic interest such as Z. Even the study of finite group representations, which probably came first historically, has become somewhat fragmented. Traditionally one considers representations of finite groups by n X n matrices with entries from C. These are the "ordinary" representations. But in the late 1930s Richard Brauer began to show the usefulness of "modular" representations (with matrix entries lying in a field of prime characteristic) as a tool in the ordinary theory and in the structure theory of finite groups. There is now an active modular industry, with a life of its own, benefiting from recent innovations such as quivers and almost split sequences in the representation theory of finite dimensional algebras (which include group algebras). Study of "integral" representations is equally active, motivated by number-theoretic considerations or by questions raised by topologists about integral group rings of fundamental groups. Received by the editors February 28,1986 and, in revised form, October 6,1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 20G40, 20C20, 17B10. Key words and phrases. Representations of finite groups, representations of algebraic groups, Steinberg representation. Research partially supported by NSF grant DMS-8502294. ©1987 American Mathematical Society 0273-0979/87 $1.00 + $.25 per page 247 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 248 J. E. HUMPHREYS The representation theory of compact groups imitates at first the finite case. It is still an essentially finite dimensional theory, with Haar integrals replacing finite sums. The main features of the ordinary representation theory of finite groups remain valid: complete reducibility, Schur's Lemma, orthogonality relations. But once one specializes to compact Lie groups, the theory (as pioneered by Elie Cartan and Hermann Weyl) takes off in new directions: highest weight of a representation, realization of irreducible representations with a fundamental dominant highest weight, Weyl character formula, in­ variant theory, unified treatment of special functions. The needs of modern physics push the subject beyond compact groups to locally compact groups, especially Lie groups: nilpotent, solvable, semisimple, reductive. But the repre­ sentation theory becomes essentially infinite dimensional and far more intri­ cate. Heavy analysis mixes with algebra in Harish-Chandra's far-reaching program, which is still at the forefront of contemporary mathematics. Similar ideas permeate the representation theory of /?-adic Lie groups. Many semisimple Lie groups are in fact linear algebraic groups (defined by polynomial equations). The finite dimensional "polynomial" representations of semisimple algebraic groups like SLM or reductive groups like GLn defined over fields of arbitrary characteristic all bear a strong family resemblance; but in prime characteristic, these usually fail to be completely reducible and take on many features of the infinite dimensional theory for Lie groups. Not content with finite dimensional groups, newer pioneers have found good reason to delve into representations of "Kac-Moody groups" associated with infinite dimensional Kac-Moody Lie algebras. The study of group representations might be regarded as yet another example of hopelessly fragmented and overspecialized mathematics, were it not for the beauty and applicability of many of its ideas. But the sheer quantity of results and techniques is a deterrent to anyone who seeks a unified overview. Lie theory provides a glimmer of hope, since there is a surprising amount of unity here—at least, after the fact. For example, Harish-Chandra's philosphy of cusp forms in the infinite dimensional setting carries over largely intact to the ordinary characters of finite groups of "Lie type" in the Dehgne-Lusztig theory. These are the groups such as VSLn(q) defined over finite fields which mimic semisimple or reductive Lie groups remarkably well and (together with alternating groups) yield most of the finite simple groups. There is also a close resemblance between the modular representation theory of these groups (in the defining characteristic) and the Cartan-Weyl theory of highest-weight represen­ tations. Just as the Kazhdan-Lusztig polynomials associated to Weyl groups have led to character formulas for infinite dimensional highest-weight modules, the analogous polynomials associated to affine Weyl groups are conjectured by Lusztig to yield character formulas for finite dimensional modular representa­ tions of groups of Lie type (and corresponding algebraic groups such as SLn). In spite of the underlying unity of much of this work, it is impossible to expose it adequately in a few pages. Instead, we focus on a single topic which conveys very well the flavor of much recent work—especially in prime char­ acteristic. Everyone who comes in contact with the representation theory of finite groups of Lie type quickly becomes aware of the Steinberg representation, a License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE STEINBERG REPRESENTATION 249 distinguished representation of prime power degree [55]. Even though it is just one of many ordinary irreducible representations (over C) or modular repre­ sentations (over fields of the prime characteristic underlying the group), it seems to turn up with disproportionate frequency in all sorts of questions: see [48] for a recent instance. Moreover, it arises independently in a number of contexts, involving not just the finite groups but also the ambient algebraic groups and their Lie algebras. The purpose of the present exposition is to sort out these facets of the Steinberg representation for the nonspecialist. Our viewpoint is to regard the Steinberg representation as a paradigm of a classical theorem due to Brauer and Nesbitt [12, Theorem 1]. a BRAUER - NESBITT THEOREM. Let G be a group of order p b9 where p is prime a and (p9b) = 1. An ordinary irreducible representation of degree divisible by p remains irreducible after "reduction modulo p", where it is also a "principal indecomposable" representation determining a "block" by itself. Moreover, the character of the representation vanishes at all elements of G having order divisible by p. To explain in detail what this theorem means and how the Steinberg representation illustrates it, we proceed in three steps. §1 describes Steinberg's construction of an ordinary representation of degree equal to the order of a Sylow /^-subgroup in a finite group of Lie type. Character values are also discussed, together with alternate approaches and applications. §2 deals with the role of the Steinberg representation as an irreducible modular representa­ tion of both the finite group and its parent algebraic group. Lie algebra representations come into the picture as well. Finally, §3 relates all of this to the projective modules in characteristic p (principal indecomposable modules) and resulting blocks (indecomposable two-sided ideals in the group algebra). The finite group G in question is realized concretely as a subgroup of an algebraic group defined over a finite field, the easiest example being SLn(g), q = power of a prime p. This particular group is of "universal" type, coming as it does from the simply connected algebraic group SLW(#), where K is an algebraic closure of the prime field ¥p. For such a group there is a single Steinberg representation. Closely related groups of Lie type such as PGLn(#), PSLM(<jr), or GLn(q) can be discussed similarly, but they may give rise to extra characters of the kind found by Steinberg (e.g., after multiplying by a power of the character det). As general background we can cite the survey article of Curtis [21] and the book of Carter [13], on representations over C, as well as the survey [37] and the forthcoming book of Jantzen [38], on modular theory. 1. St as an ordinary representation. 1.1 Steinberg's construction. The existence of characters of prime power degree for finite groups of Lie type was perhaps first observed around 1900, in the work of Frobenius and Schur on characters of SL2(q). But not until the 1950s did such observations become systematic, in the early work of Steinberg on classical groups (cf. [54] and part I of [55]) and the independent work of Green on GLn(q). After the publication of Chevalley's famous Tôhoku paper on finite simple groups in 1955, Steinberg, Tits, and others rapidly completed License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 250 J. E. HUMPHREYS the Hst of simple groups of Lie type and devised axiomatic descriptions to facilitate their further study. In part II of [55], Steinberg constructed what is now called the Steinberg representation for the Chevalley groups.
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